June 2012 “Understanding is a measure of the quality and quantity of connections that an idea has with existing ideas.” (VandeWalle, Pg. 23)

1. Conceptual understanding 2. Procedural fluency 3. Strategic competence 4. Adaptive reasoning 5. Productive disposition Source: Common Core (Pg. 6)

 According to a 2007 Trends in International Mathematics and Science Study, Singapore students are among the best in the world in math achievement.  The Singapore model drawing approach bridges the gap between the concrete and abstract models we tend to jump to in the US.  Model drawing reinforces students’ visualization and understanding of math processes.  Model drawing can be used effectively to solve 80 percent of problems in all texts.

 Computation is about students comprehending what they are doing, not following a set of rules.  Students need to understand both what to do and why.  Students will have a variety of strategies to solve problems.  Changes students attitudes toward math and problem solving.  Language based learning- think alouds and math talks  Can be used as a supplement to an adopted curriculum  Singapore Math Video Singapore Math Video

 Instruction begins at the concrete level with manipulatives to build understanding of basic concepts and skills. (begin with proportional manipulatives)  Then students are introduced to the pictorial stage: model drawing.  Students are not introduced to formulaic or algorithmic procedures, the abstract, until they have mastered model-drawing.

Discuss. Catch thinking on chart paper. Write a summary paragraph.

1) Teacher modeling and thinking aloud about the strategy 2) Students practice with the teacher 3) Students practice in small groups 4) Students practice in partners 5) Independent practice

Definition: Number sense is a “good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Van de Walle, Pg. 119)

 “I’m thinking…” “I’m wondering…” “What are you thinking?” “How did you figure that out?” “Is there another way?” “Why did you choose this way?” “How do you know this answer is correct?” “What would happen if?”  Model clear, explicit language about concepts  Mathematical thinking and language promote more understanding than memorization or rules

 =  First, I’m going to add the hundreds. That means =700. Now I will add the tens, four tens (40) plus seven tens (70) equals eleven tens. I can make 110. So now I have Now I will add the ones and 6+5 makes 11. This is one ten and one 1 so I have My answer is 821.  Place Value Talk is critical!

Purpose: Modeling, communicating, promoting a more efficient strategy, promoting reasoning, moving to a more sophisticated level of thinking.

Purpose: “Unstick” someone, get help, clarify, promote deeper thinking, make connections

Purpose: Promote productive math conversations Example: Practicing questions in multiple- choice format  Step 1: Solve individually. Write down your answer.  Step 2: Compare. Same or different?  Step 3: Explain why you chose that answer.

Purpose: Use words that proficient mathematicians would use, make connections Example: End of Year Jeopardy Review Game.  500 point question: What is addition?

 Look at their work. Do the model, the picture, and the equation match the question and each other?  Read or listen to their explanation.  Ask a math question.  Seek professional help – Ask a student expert, the teacher, or other adult.

 Place Value Strips  Place Value Disks  Place Value Chart  Number-bond cards  Part-whole cards  Decimal Tiles  Decimal Strips  Gratiot Isabella ISD Maniplatives Link Gratiot Isabella ISD Maniplatives Link

 Begin with no grouping  Sequence 1. Number bonds 2. Decomposing numbers 3. Left-to-right addition 4. Place value disks and charts 5. Vertical addition 6. Traditional addition

 Spatial Relationships  pattered arrangements  One and Two More, One and Two Less  counting on and counting back  7 is 1 more than 6 and it is 2 less than 9  Anchoring Numbers to 5 and 10  using 5 and 10 to build on and break from is foundational for working with facts  Part-Part-Whole Relationships  understand that a number is can be made of 2 or more parts

 A Pre-Place Value Relationship with 10  11 through 20, think 10 and some more  Extending More and Less Relationships  i.e. 17 is one less than 18 like 7 is one less than 8  Double and Near-Double Relationships  special cases of the part-part-whole construct  use pictures

Build Up and Down through 10 Break Apart to Find an Unknown Fact

 Estimation and Measurement  More or less than _______?  Closer to _____ or _____?  About _______.  More Connections  Add a Unit to Your Number  Is It Reasonable?  Graphs  Make bar graphs and pictographs

Big Ideas: 1. Number relationships provide the foundation for strategies that help children remember the basic facts. (i.e. relate to 5, 10, and doubles…) 2. “Think addition” is the most powerful way to think of subtraction facts. 3. All of the facts are conceptually related. You can figure out new or unknown facts from those you already know. 4. What is mastery? 3 seconds or less

 33+56=  Decompose each number by place value.  Put the tens together and one ones together.  (30+50) + (3+6)  =

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 Add from left to right.

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 Begin with no regrouping  Sequence 1. Number Bonds 2. Place Value Disks and Charts 3. Traditional Subtraction

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 86-8  = 78  86-8=

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 Before “memorizing” multiplication facts, students must first understand the concept of multiplication----they must know it is repeated addition with special attention to place value  Stages 1. Number bonds 2. Place value disks and charts 3. The distributive property 4. Area model 5. Traditional multiplication

 or Vocab: Factor x Factor=Product

 141x

 141x

 141x x 3=

 45x3 (40x3) + (5x3) = 135

 6x x33= =198

 15x x26=( )+(60+30) =390 15x26=390

 Begin by introducing division as repeated subtraction  Use number bonds to demonstrate the inverse relationship of multiplication and division  Sequence for teaching 1. Number bonds 2. Place value disks and charts 3. The distributive property 4. Partial quotient division 5. Traditional long division 6. Short division

9  27 ? What should the second factor be? Vocab: Dividend ÷ Divisor=Quotient

 64÷5  Step 1 Build with disks

 64÷5  Step 2: The divisor tells us how many groups we need: Draw 5 rows  1  2  3  4 

 64÷5  Step 3: Begin with the tens. Do we have enough tens to put one in every row? Yes  1  2  3  4 

 64÷5  Step 4: If there are any large disks left, trade them for an equivalent value of smaller disks  1  2  3  4 

 64÷5  Step 5:Divide the ones equally among the groups  1  2  3  4  =12 64÷5=12 r

 345÷3=  345= unfriendly 345= (300÷3) + (30÷3) + (15÷3)= = 115

 =126

157 

Purpose: Provide context, aid concept development, make real-world connections Activity: Article: Round-Robin Story TellingRound-Robin Story Telling

 Author(s): Joseph Martinez and Nancy Martinez  Source: Instructor (1990) (Mar. 2000): p70.  Round-robin storytelling works well as a math activity with elementary-school students. Provide each small group with a story beginning such as: Jon and Michelle make paper hats. Their standard hat is paper with a tissue tassel. It costs $.30 to make and sells for $1.25. Within each group, students add to the story For example: They also make a fancy hat that sells for $2.50. The last student in the group wraps up the story and poses the problem: Jon and Michelle sell 60 standard hats, and 40 fancy hats. How much profit will they make? Each group works to solve its problem. The groups then swap stories.

 Not done yet

Purpose: Concept development, communicating ideas Activity: Work with a partner or in a small group. Choose one of the problems that we wrapped a story around. 1- Build a Model 2- Draw a Picture 3- Write an Equation 4- Write your answer in a complete sentence. 5- Explain.

Purpose: Assure progress in terms of academic achievement Activity:

 Why Before How: Singapore Math Computation Strategies: Jana Hazekamp